We estimated the equilibrium OMAD [operating (profit) margin after depreciation] of certain U.S. retailers using an autoregressive (AR(1)) model built-in RoyaltyStat. RoyaltyStat uses the Gauss run-time engine, so the regression estimates are reliable.
We used an AR(1) model of company OMAD (which we call Y):
(1) Y(t + 1) = α + β Y(t) + U(t),
where Y(t) is stable when the absolute value of the slope coefficient is lower than one (|β| < 1). Among others, Walter Enders, Applied Econometric Time Series, John Wiley & Sons, 1995, is a useful reference.
The error term U(t) has zero-mean and a constant variance, σ2, and year includes all the data available for the period t = 1, 2, 3, …, T. We have criticized the OECD TNMM convention that T = 3 years, including the current audit year and two prior years. We get more reliable results using as many years of OMAD data as available, except that we should exclude outliers because they distort the point estimates and their standard errors.
If μ is the mean (average) OMAD, and the operator E is expected value, we can obtain the fixed-point equilibrium for this pervasive profit indicator:
(2) E(Y(t + 1)) = α + β E(Y(t)), where E(U(t)) = 0.
(3) μ = α + β μ from which we get the fixed-point equilibrium:
(4) μ = α / (1 − β).
If the intercept α = 0, then μ = 0 and Y(t) fluctuates around zero.
The intercept represents the routine OMAD expected to be earned by every company in the industry; the slope coefficient is the non-routine OMAD multiplier. In economics, the non-routine return is attibuted to economic concentation (measured by significant market-share) in competition policy, intangibles in transfer pricing, and significant risk in the financial literature. For application of AR(1) to historical company data in various countries, see Dennis Mueller (ed.), The Dynamics of Company Profits, Cambridge University Press, 1990.
Circa 1990, we wrote a memorandum to Charles Triplett, then Deputy Associate Chief Counsel (International), during my early participation in the drafting of the U.S. transfer pricing regulations, expounding this AR(1) model as a statistical method to obtain stable company OMAD among the selected comparables; however, he and other IRS tax counsel involved in the drafting of the U.S. (1992, 1993 and 1994) transfer pricing regulations found this AR approach recondite, and thus this economic pregnant idea was abondoned. However, we can utilize this AR(1) model given the reliability conditions required under 26 CFR §1.482-1(e)(2)(iii)(B), which provides: “The interquartile range ordinarily provides an acceptable measure of this range; however a different statistical method may be applied if it provides a more reliable measure [of the arm's length range].”
Suppose that a U.S. controlled retailer (tested party) under audit has 13 comparables selected from a list of 21 retailers. The table below summarizes our empirical results of the equilibrium OMAD computed using equation (4). GVKEY is Standard & Poor's Compustat “Global Vantage Key” that is unique for every company. Count (sample size) is the number of annual OMAD terminating in 2016, going back various years as the data are available per company:
Equilibrium Point OMAD (%)
St. Error of Point Estimate (%)
Regression Std. Error (%)
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The standard error of the equilibrium OMAD is small compared to the equilibrium OMAD, suggesting that this AR(1) model has a major advantage in producing low coefficients of variation. E.g., counting 33 years of data, Best Buy has an equilibrium OMAD = 3.9% ± 1.1%. Low coefficients of variation imply reliable measures of the variable (OMAD) of interest, because the reliability of the estimated mean or expected value depends on the standard deviation. The coefficient of variation is the ratio of the standard deviation to the mean and measures the relative reliability of the sample mean. See Joseph Gastwirth, Statistical Reasoning in Law and Public Policy, Academic Press, 1988, Vol. 2, pp. 493-494.
Similar to the derivation of the equilibrium point, we derive the standard error of the equilibrium position by letting V denote the variance operator, and then assuming a constant or stable variance over time:
(6) V(Y(t + 1)) = V(Y(t )) = Σ.
Applying this time-invariant measure of data spread (Σ) to equation (1), we obtain:
(7) Σ = β2 Σ + σ2, which yields the equilibrium variance:
(8) Σ = σ2 / (1 − β2) for β < 1.
The standard error of the equilibrium position (4) is the square root of (8). We know from first-order difference equations, such as AR(1), that we get stable results when β < 1. Else, if β ≥ 1, the variance grows exponentially with time, and there is no finite limit for long horizons, i.e., we get no fixed-point equilibrium value.
A principal conclusion that we draw is that an AR(1) model can produce reliable long-run estimates of OMAD, and we can compare the results of the AR(1) model with the naive equation applied to the rote three years of data provided in the OECD Transfer Pricing Guidelines to determine what's best under the facts and circumstances. Naivete is not a good saddle to ride in a controversial arena such as arm's length income tax determination.
Ednaldo Silva (Ph.D.) is founder and managing director at RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: firstname.lastname@example.org
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